This paper presents a combination of searches for the nonresonant production of Higgs boson pairs (HH) in proton-proton collisions at a centre-of-mass energy of 13 TeV. The data set was collected by the CMS experiment at the LHC from 2016 to 2018 and corresponds to a total integrated luminosity of 138 fb$^{-1}$. The observed (expected) upper limit on the inclusive HH production cross section relative to the standard model (SM) prediction is found to be 3.5 (2.5). Assuming all other Higgs boson couplings are equal to their SM values, the Higgs boson trilinear self-coupling modifier $κ_λ=λ_3/λ_{3}^\text{SM}$ is constrained in the range $-$1.35 $\leq$ $κ_λ$ $\leq$ 6.37 at 95% confidence level. Similarly, for the coupling modifier $κ_{2\mathrm{V}}$, which governs the interaction between two vector bosons and two Higgs bosons, we have excluded $κ_{2\mathrm{V}}$ = 0 at more than 5 standard deviations for all values of $κ_λ$. At 95% confidence level assuming other couplings are equal to their SM values, $κ_{2\mathrm{V}}$ is constrained in the range 0.64 $\leq$ $κ_{2\mathrm{V}}$ $\leq$ 1.40. This work also studies HH production in several new physics scenarios, using the Higgs effective field theory (HEFT) framework. The HEFT framework is further exploited to study various ultraviolet complete models with an extended Higgs sector and set constraints on specific parameters. An extrapolation of the results to the integrated luminosity expected after the high-luminosity upgrade of the LHC is reported as well.